Example: The orthant already satisfies homogeneity

For $m\ge 3$, $K={{\mathbb{R}}_{\ge 0}}^m$ is proper and homogeneous: for $u,v\in \mathrm{int}K$, the diagonal map

is an automorphism and sends $u$ to $v$. It fails strict convexity because $e_1,e_2$ are non-proportional elements of $K$, while $e_1+e_2\in \partial K$. Thus H-homogeneity alone does not exclude the orthant countermodel.